🧩 Fractions and Decimals in Context

Part 1 of 7 — Fractions and Decimals on the AR Subtest

Topic Overview

What These Questions Look Like

The Arithmetic Reasoning (AR) subtest is officially described as the "ability to solve basic arithmetic word problems," and fractions and decimals show up in a big share of them. They rarely say "compute $\frac{3}{4} \times 36$." Instead they tell a short story:

• A fraction of a group: "Three-fourths of the 36 soldiers in a unit passed the swim test..."
• Measurements in mixed numbers: board lengths, hours worked, miles run, cups in a recipe.
• Money with decimals: a paycheck at $14.50 an hour, the change from a $20 bill, the cost of 12 gallons of gas.

What is really being tested is whether you can turn the words into the right operation and carry it out cleanly by hand.

The Conditions You Work Under

If you take a paper version of the test, the pace is much faster, so the shortcuts in this lesson matter even more.

AR is one of the four subtests in your AFQT score, the score that decides whether you can enlist, and it also feeds job line scores such as the Army GT score. Fraction and decimal points are some of the most winnable points on it.

🧩 Fractions and Decimals in Context

Part 2 of 7 — Finding a Fraction of a Quantity

"Of" Means Multiply

"Five-sixths of 42" means $\frac{5}{6} \times 42$. Without a calculator, do it in two easy steps:

1. Divide by the denominator (the bottom number) to find the size of one equal part.
2. Multiply by the numerator (the top number) to count how many parts you need.

Example 1. What is $\frac{5}{6}$ of 42?

$42 \div 6 = 7 \qquad 7 \times 5 = 35$

Estimate: $\frac{5}{6}$ is a little less than the whole, so the answer should be a little less than 42. $35$ fits.

Example 2 — a fraction of a population. A town has 2,400 residents, and $\frac{3}{8}$ of them are under 25 years old. How many residents are under 25?

$2{,}400 \div 8 = 300 \qquad 300 \times 3 = 900$

Estimate: $\frac{3}{8}$ is a bit less than half, and half of 2,400 is 1,200. So 900 residents is reasonable.

Key Fact: Divide first, then multiply. Dividing first keeps the numbers small, and in AR problems the division almost always comes out even.

🧩 Fractions and Decimals in Context

Part 3 of 7 — Adding and Subtracting Fractions and Mixed Numbers

Common Denominators First

You can only add or subtract fractions when they count the same size pieces. Thirds and fourths are different sizes, so rewrite both as twelfths first.

Example 1. A cook uses $\frac{1}{3}$ cup of oil in the batter and $\frac{1}{4}$ cup in the pan. How much oil is that altogether?

The smallest number both 3 and 4 divide into is 12:

$\frac{1}{3} = \frac{4}{12} \qquad \frac{1}{4} = \frac{3}{12} \qquad \frac{4}{12} + \frac{3}{12} = \frac{7}{12} \text{ cup}$

Add the numerators; keep the denominator.

Never add the tops and the bottoms. $\frac{1}{3} + \frac{1}{4}$ is not $\frac{2}{7}$. A quick check shows why: $\frac{2}{7}$ is less than $\frac{1}{3}$ by itself, and adding more oil cannot leave you with less.

Key Fact: If you cannot spot the common denominator, multiply the two denominators. It always works ($3 \times 4 = 12$); you may just need to simplify at the end.

🧩 Fractions and Decimals in Context

Part 4 of 7 — Multiplying and Dividing Fractions in Context

Multiplying: Tops Times Tops, Bottoms Times Bottoms

To multiply fractions, multiply the numerators and multiply the denominators. No common denominator needed.

Example 1. Two-thirds of a field is planted, and $\frac{3}{4}$ of the planted part is corn. What fraction of the whole field is corn?

"Three-fourths of two-thirds" means multiply:

$\frac{3}{4} \times \frac{2}{3}$

Cancel first. The 3 on top cancels the 3 on the bottom, and the 2 on top goes into the 4 on the bottom twice, leaving a 2 on the bottom:

$\frac{3}{4} \times \frac{2}{3} = \frac{1}{2} \times \frac{1}{1} = \frac{1}{2}$

(Without canceling: $\frac{6}{12}$, which simplifies to the same $\frac{1}{2}$.)

Half of the field is corn. Reasonableness: a fraction of a fraction must be smaller than both, and $\frac{1}{2}$ is smaller than $\frac{2}{3}$ and $\frac{3}{4}$. ✓

Key Fact: Cancel any top number with any bottom number before you multiply. The numbers stay small and you rarely need to simplify at the end.

🧩 Fractions and Decimals in Context

Part 5 of 7 — Decimals and Money by Hand

Money problems are decimal problems, and on AR you do them without a calculator. Each operation has one rule about the decimal point.

Adding and Subtracting: Line Up the Decimal Points

Stack the numbers so the decimal points sit in one column, and fill empty places with zeros.

Example 1 — making change. A soldier buys a meal for $13.47 and pays with a $20 bill. How much change does he get?

So the change is $6.53.

Count-up check (the way a cashier does it): from $13.47, add 3 cents to reach $13.50, 50 cents to reach $14.00, then $6 to reach $20.00. Total: $0.03 + 0.50 + 6 = 6.53$, so $6.53. ✓

Key Fact: Writing $20 as 20.00 before you subtract prevents the most common change-making slip: forgetting to borrow, which gives $7.53.

🧩 Fractions and Decimals in Context

Part 6 of 7 — Test Strategy and Common Traps

The wrong answer choices on AR are not random. Each one is the result of a specific, common mistake. If you know the mistakes, you can see the traps coming.

The Six Big Traps

Key Fact: Each of these traps produces a number that appears among the answer choices. Finding your answer in the list does not prove it is right. Your check does.

🧩 Fractions and Decimals in Context

Part 7 of 7 — Mixed Mastery Set

This set mixes everything from Parts 2 through 5 the way the real AR subtest does: a few quick ones, several medium ones, and a few multi-step problems. Treat it like test day:

• No calculator. Use scratch paper for every step.
• Read the last sentence first and circle words like left, not and rest.
• Estimate before you compute, and check before you answer, because on the real test you cannot go back.
• Aim for about 3.5 minutes per question or less.